ON 1-DEPENDENT PROCESSES AND K-BLOCK FACTORS

A stationary process {X(n)}(n epsilon Z) is said to be k-dependent if {X(n)}(n<0) is independent of {X(n)}(n>k-1). It is said to be a k-bloc k factor of a process {Y-n} if it can be represented as X(n) = f(Y-n,. ..,Y-n+k-1), where f is a measurable function of k variables. Any (k 1)-block factor of an i.i.d. process is k-dependent. We answer an old question by showing that there exists a one-dependent process which i s not a k-block factor of any i.i.d. process for any k. Our method als o leads to generalizations of this result and to a simple construction of an eight-state one-dependent Markov chain which is not a two-block factor of an i.i.d. process.